Dimensionless variables

for the channel electrode

In this appendix it is shown how the material balance equation may be reduced into a dimensionless form. A full treatment leads to the result that the dimensionless current may be expressed as a sole function of two mass transport parameters, a dimensionless time parameter and a number of dimensionless rate constants. If the Lévêque approximation is used to linearise the convective velocity profile, the two mass transport parameters reduce to a single one - the shear rate Peclet number.

For a transport-limited electrolysis (for example a one electron reduction):

A + e => B | (A1.1) |

at a channel electrode the steady-state convective-diffusion equation is:

= 0 | (A1.2) |

If the dimensionless variables are introduced:

, | (A1.3) |

the convective-diffusion equation becomes:

(A1.4) |

Under laminar flow conditions, when a Poiseille flow profile has fully developed, v_{x} is given by:

(A1.5) |

(A1.6) |

where:

and | (A1.7) |

Hence the concentration is a function of four parameters:

a = a(c,y,p_{1},p_{2}) | (A1.8) |

The dimensionless current (Nusselt number) is given by:

(A1.9) |

hence:

(A1.10) |

c is set to zero and y disappears once the integral is evaluated, therefore the Nusselt number is purely a function of p_{1} and p_{2}.

For an irreversible transport-limited ECE or EC_{2}E reaction (written as a reduction):

A + e^{-} = BnB => C C + e ^{-} = Products | (A1.11) |

(where n=1 defines an ECE reaction and n=2 defines an EC_{2}E reaction), the steady-state material balance equations are:

= 0 | (A1.12) |

= 0 | (A1.13) |

= 0 | (A1.14) |

where k_{c} is given by:

(A1.15) |

These can be reduced into dimensionless forms in the same way. Species A does not have any kinetic terms in its material balance equation. The equation for species B becomes:

(A1.16) |

The kinetic term is known as the dimensionless rate constant:

(A1.17) |

Hence

b = b(c,y,p_{1},p_{2},K_{norm}) | (A1.18) |

Similarly the equation for species C becomes:

(A1.19) |

Hence

c = c(c,y,p_{1},p_{2},K_{norm},b) = c(c,y,p_{1},p_{2},K_{norm}) | (A1.20) |

Only species A and C are electroactive, so the dimensionless current is given by:

(A1.21) |

The first integral has been given above and if the second is treated analogously:

(A1.22) |

Where f and f' are used to represent different functions. Hence the ratio of the ECE current to the electron transfer current:

(A1.23) |

(A1.24) |

(A1.25) |

(A1.26) |

(A1.27) |

If the Lévêque approximation is made:

(A1.28) |

the dimensionless mass transport equation reduces to:

(A1.29) |

(A1.30) |

thus:

a = a(c,y,P_{s}) | (A1.31) |

and Nu simplifies to a unique function of the shear Peclet number.

If we redefine the dimensionless variable in the y co-ordinate:

(A1.32) |

the mass transport equation becomes:

(A1.33) |

which simplifies to:

(A1.34) |

where:

p = P_{s}^{-2/3} | (A1.35) |

If axial diffusion can be ignored, the equation reduces to:

(A1.36) |

and the dependence on p is lost. The Nusselt number is a constant (from the integral) multiplied by P_{s1/3}_{.}

(A1.37) |

The constant may be found by solving equation (A1.37) analytically, giving the classical Levich equation^{1} :

(A1.38) |

^{1}V.G. Levich,*Physicochemical Hydrodynamics*, Prentice-Hall, Engelwood Cliffs, New Jersey, (1962), p112.